A hidden Markov model (HMM) was traditionally used to model one dimensional data for the purposes of pattern recognition. For example, HMMs have been widely used in speech recognition. Speech signals are inherently one dimensional. In speech processing, HMM can be used to model phonemes, words, or even phrases. One of the important characteristics of HMM is its ability to cope with variations in time and in feature space simultaneously, allowing to model data with variations along different dimensions. For example, different people may speak English with different accents. A particular person may speak with different acoustic features at different times. When HMM models are trained based on speech from different speakers at different times, derived HMM models may capture the intrinsic features of the speech and provide models that can tolerate different degrees of variation.
Many pattern recognition problems arise from data residing in a space of higher dimension. For example, identifying a specific object from an image may be a pattern recognition task performed in a two-dimensional (at least) feature space. Detecting the regularity of the heart beat rhythm of a patient from a series of three dimensional heart images reconstructed based on computerized tomography (CT) scans over time may be a pattern recognition problem in a four dimensional space.
Some efforts have been made to extend one-dimensional HMM to more complex structures. For example, a coupled hidden Markov model (CHMM) has been proposed to model two-dimensional data. In CHMM, two one-dimensional HMMs are coupled so that states from one HMM can transit to states of the other HMM. Transition probabilities among states are computed in view of the two-dimensional configuration of the HMMs.